CUET UG 2025 — Mathematics Algebra

The cofactor $C_{ij}$Cij​ is given by $(-1)^{i+j} M_{ij}$(−1)i+jMij​, where $M_{ij}$Mij​ is the determinant of the submatrix after removing the $i$i-th row and $j$j-th column.

The matrix is:
$$A = \begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 0 \\ 4 & 1 & 5 \end{bmatrix}$$A=​214​−121​305​​

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$$C_{23} = (-1)^{2+3} \begin{vmatrix} 2 & -1 \\ 4 & 1 \end{vmatrix}$$C23​=(−1)2+3​24​−11​​

$$C_{23} = -1(2(1) - (-1)(4))$$C23​=−1(2(1)−(−1)(4))

$$C_{23} = -1(2 + 4)$$C23​=−1(2+4)

$$C_{23} = -6$$C23​=−6

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$$C_{31} = (-1)^{3+1} \begin{vmatrix} -1 & 3 \\ 2 & 0 \end{vmatrix}$$C31​=(−1)3+1​−12​30​​

$$C_{31} = 1((-1)(0) - (3)(2))$$C31​=1((−1)(0)−(3)(2))

$$C_{31} = 1(0 - 6)$$C31​=1(0−6)

$$C_{31} = -6$$C31​=−6

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$$C_{22} = (-1)^{2+2} \begin{vmatrix} 2 & 3 \\ 4 & 5 \end{vmatrix}$$C22​=(−1)2+2​24​35​​

$$C_{22} = 1(2(5) - (3)(4))$$C22​=1(2(5)−(3)(4))

$$C_{22} = 1(10 - 12)$$C22​=1(10−12)

$$C_{22} = -2$$C22​=−2

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$$C_{23} + C_{31} - C_{22} = -6 + (-6) - (-2)$$C23​+C31​−C22​=−6+(−6)−(−2)

$$= -6 - 6 + 2$$=−6−6+2

$$= -10$$=−10